Abstract

In this paper, we introduced two new classes of nonlinear mappings in Hilbert spaces. These two classes of nonlinear mappings contain some important classes of nonlinear mappings, like nonexpansive mappings and nonspreading mappings. We prove fixed point theorems, ergodic theorems, demiclosed principles, and Ray's type theorem for these nonlinear mappings.

Next, we prove weak convergence theorems for Moudafi's iteration process for these nonlinear mappings. Finally, we give some important examples for these new nonlinear mappings.

Keywords

1 Introduction

Let H be a real Hilbert space, and let C be a nonempty closed convex subset of H. Then, a mapping T : C → C is said to be nonexpansive if ||Tx - Ty|| ≤ ||x - y|| for all x, y∈C. The set of fixed points of T is denoted by F (T). The class of nonexpansive mappings is important, and there are many well-known results in the literatures. From literatures, we observe the following fixed point theorems for nonexpansive mappings in Hilbert spaces.

In 1965, Browder [1] gave the following demiclosed principle for nonexpansive mappings in Hilbert spaces.

Theorem 1.1. [1] Let C be a nonempty closed convex subset of a real Hilbert space H. Let T be a nonexpansive mapping of C into itself, and let {xn } be a sequence in C. If xn⇀w and limn→∞||xn-Txn||=0, then Tw = w.

In 1971, Pazy [2] gave the following fixed point theorems for nonexpansive mappings in Hilbert spaces.

Theorem 1.2. [2] Let H be a Hilbert space and let C be a nonempty closed convex subset of H. Let T : C → C be a nonexpansive mapping. Then, {Tnx} is a bounded sequence for some x∈C if and only if F (T) ≠ ∅.

In 1975, Baillon [3] gave the following nonlinear ergodic theorem in a Hilbert space.

Theorem 1.4. [4] Let C be a nonempty closed convex subset of a real Hilbert space H. Then, the following conditions are equivalent.

(i)

Every nonexpansive mapping of C into itself has a fixed point in C;

(ii)

C is bounded.

On the other hand, a mapping T : C → C is said to be firmly nonexpansive [5]

if

||Tx-Ty||2≤⟨x-y,Tx-Ty⟩

for all x, y∈C, and it is an important example of nonexpansive mappings in a Hilbert space.

In 2008, Kohsaka and Takahashi [6] introduced nonspreading mapping and obtained a fixed point theorem for a single nonspreading mapping and a common fixed point theorem for a commutative family of nonspreading mappings in Banach spaces. A mapping T : C → C is called nonspreading [6] if

Remark 1.2. The class of asymptotic TJ mappings contains the class of TJ-1 mappings and the class of nonexpansive mappings in a Hilbert space. Indeed, in Definition 1.2, we know that

(i)

if α (x) = 2 and β(x) = 0 for each x∈C, then T is a nonexpansive mapping;

(ii)

if α(x) = β(x) = 1 for each x∈C, then T is a TJ-1 mapping.

On the other hand, the following iteration process is known as Mann's type iteration process [10] which is defined as

xn+1=αnxn+(1-αn)Txn,n≥0,

where the initial guess x0 is taken in C arbitrarily and the sequence {αn } is in the interval [0, 1].

In 2007, Moudafi [11] studied weak convergence theorems for two nonexpansive mappings T1, T2 of C into itself, where C is a closed convex subset of a Hilbert space H. They considered the following iterative process:

x0∈C chosen arbitrarily,yn=βnT1xn+(1-βn)T2xnxn+1=αnxn+(1-αn)yn

for all n∈N, where {αn } and {βn } are sequences in [0, 1] and F(T1) ∩ F(T2) ≠ ∅. In 2009, Iemoto and Takahashi [8] also considered this iterative procedure for T1 is a nonexpansive mapping and T2 is nonspreading mapping of C into itself.

Motivated by the works in [8, 11], we also consider this iterative process for asymptotic nonspreading mappings and asymptotic TJ mappings.

2 Preliminaries

Throughout this paper, let ℕ be the set of positive integers and let ℝ be the set of real numbers. Let H be a (real) Hilbert space with inner product 〈·, ·〉 and norm || · ||, respectively. We denote the strongly convergence and the weak convergence of {xn } to x∈H by xn → x and xn⇀x, respectively. From [12], for each x, y∈H and λ∈ 0[1], we have

||λx+(1-λ)y||2=λ||x||2+(1-λ)||y||2-λ(1-λ)||x-y||2.

Let ℓ∞ be the Banach space of bounded sequences with the supremum norm. A linear functional μ on ℓ∞ is called a mean if μ(e) = || μ || = 1, where e = (1, 1, 1, ....). For x = (x1, x2, x3, ....), the value μ(x) is also denoted by μn (xn ). A Banach limit on ℓ∞ is an invariant mean, that is, μn (xn ) = μn (xn+1). If μ is a Banach limit on ℓ∞, then for x = (x1, x2, x3, ...) ∈ℓ∞,

liminfn→∞xn≤μnxn≤limsupn→∞xn.

In particular, if x = (x1, x2, x3, ...) ∈ℓ∞ and xn → a∈ℝ, then we have μ(x) = μnxn = a. For details, we can refer [13].

Let C be a nonempty closed convex subset of a real Hilbert space H, and let T : C → C be a mapping, and let F (T) denote the set of fixed points of T. A mapping T : C → C with F (T) ≠ ∅ is called quasi-nonexpansive if ||x - Ty|| ≤ ||x - y|| for all x∈F (T) and y∈C. It is well known that the set F (T) of fixed points of a quasi-nonexpansive mapping T is a closed and convex set [14]. Hence, if T : C → C is an asymptotic nonspreading mapping (resp., asymptotic TJ mapping) with F (T) ≠ ∅, then T is a quasi-nonexpansive mapping and this implies that F (T) is a nonempty closed convex subset of C.

Proposition 2.1. Let C be a nonempty closed convex subset of a Hilbert space H. Let α, β be the same as in Definition 1.1. Then, T : C → C is an asymptotic nonspreading mapping if and only if

Lemma 2.4. [9] Let H be a Hilbert space, let C be a nonempty closed convex subset of H, and let T be a mapping of C into itself. Suppose that there exists an element x∈C such that {Tnx} is bounded and

μn||Tnx-Ty||2≤μn||Tnx-y||2,∀y∈C

for some Banach limit μ. Then, T has a fixed point in C.

3 Main results

In this section, we study the fixed point theorems, ergodic theorems, demiclosed principles, and Ray's type theorems for asymptotic nonspreading mappings and for asymptotic TJ mappings in Hilbert spaces.

Theorem 3.2 generalizes Theorem 1.2 since the class of asymptotic TJ mappings contains the class of nonexpansive mappings. By Theorems 3.1 and 3.2, we also get the following result as special cases, respectively.

Theorem 3.5 generalizes Theorem 1.1 since the class of asymptotic TJ mappings contains the class of nonexpansive mappings. Furthermore, we have the following results as special cases of Theorems 3.4 and 3.5, respectively.

Corollary 3.5. [8] Let C be a nonempty closed convex subset of a real Hilbert space H. Let T be a nonspreading mapping of C into itself, and let {xn } be a sequence in C. If xn⇀w and limn→∞||xn-Txn||=0, then Tw = w.

Corollary 3.6. [9] Let C be a nonempty closed convex subset of a real Hilbert space H. Let T be a TJ-1 mapping of C into itself, and let {xn } be a sequence in C. If xn⇀w and limn→∞||xn-Txn||=0, then Tw = w.

3.3: Ergodic theorems

Theorem 3.6. Let C be a nonempty closed convex subset of a real Hilbert space H, and let T : C → C be an asymptotic nonspreading mapping. Let α and β be the same as in Definition 1.1. Suppose that α(x)/β(x) = r > 0 for all x∈C. Then, the following conditions are equivalent.

(i)

F (T) ≠ ∅;

(ii)

for any x∈C, Snx=1n∑k=0n-1Tkx converges weakly to an element in C.

In fact, if F (T) ≠ ∅, then Snx⇀limn→∞PTnx for each x∈C, where P is the metric projection of H onto F (T).

Proof. (ii)) ⇒ (i): Take any x∈C and let x be fixed. Then, Snx⇀v for some v∈C. Then, v∈F (T). Indeed, for any y∈C and k∈ℕ, we have

Since y is any point of C and r > 0, let y = v and this implies that Tv = v.

(i)⇒ (ii): Take any x∈C and u∈F (T), and let x and u be fixed. Since T is an asymptotic nonspreading mapping, ||Tnx - u|| ≤ ||Tn-1x - u|| for each n∈ℕ. By Lemma 2.2, {PTnx} converges strongly to an element p in F (T). Then for each n∈ℕ,

||Snx-u||≤1n∑k=0n-1||Tkx-u||≤||x-u||.

So, {Snx} is a bounded sequence. Hence, there exists a subsequence {Snix} of {Snx} and v∈C such that Snix⇀v. As the above proof, Tv = v.

By Lemma 2.1, for each k∈ℕ, 〈Tkx - PTkx, PTkx - u〉 ≥ 0. And this implies that

Adding these inequalities from k = 0 to k = n - 1 and dividing n, we have

Snx-1n∑k=0n-1PTkx,u-p≤||x-p||n∑k=0n-1||PTkx-p||.

Since Snix⇀v and PTkx → p, we get 〈v - p, u - p〉 ≤ 0. Since u is any point of F (T), we know that v = p.

Furthermore, if {Snjx} is a subsequence of {Snx} and Snj⇀w, then w = p by following the same argument as in the above proof. Therefore, Snx⇀p=limn→∞PTnx, and the proof is completed. □

Theorem 3.7. Let C be a nonempty closed convex subset of a real Hilbert space H, and let T : C → C be an asymptotic TJ mapping. Let α and β be the same as in Definition 1.2. Suppose that β(x)/α(x) = r > 0 for all x∈C. Then, the following conditions are equivalent.

(i)

F (T) ≠ ∅;

(ii)

for any x∈C, Snx=1n∑k=0n-1Tkx converges weakly to an element in C.

In fact, if F (T) ≠ ∅, then Snx⇀limn→∞PTnx for each x∈C, where P is the metric projection of H onto F (T).

Proof. The proof of Theorem 3.7 is similar to the proof of Theorem 3.6, and we only need to show the following result.

Take any x∈C and let x be fixed. Then, Snx⇀v for some v∈C. Then, v∈F (T). Indeed, for any y∈C and k∈ℕ, we have

And following the same argument as the proof of Theorem 3.6, we get Theorem 3.7. □

By Theorems 3.6 and 3.7, we get the following result.

Corollary 3.7. [9, 16] Let C be a nonempty closed convex subset of a real Hilbert space H, and let T : C → C be any one of nonspreading mapping, TJ-1 mapping, and TJ-2 mapping. Then, the following conditions are equivalent.

(i)

F (T) ≠ ∅;

(ii)

for any x∈C, Snx=1n∑k=0n-1Tkx converges weakly to an element in C.

In fact, if F (T) ≠ ∅, then Snx⇀limn→∞PTnx for each x∈C, where P is the metric projection of H onto F (T).

3.4 Ray's type theorems

Theorem 3.8. Let C be a nonempty closed convex subset of a real Hilbert space H. Then, the following conditions are equivalent.

(i)

Every asymptotic TJ mapping of C into itself has a fixed point in C;

(ii)

C is bounded.

Proof. (i)⇒ (ii): Suppose that every asymptotic TJ mapping of C into itself has a fixed point in C. Since the class of asymptotic TJ mappings contains the class of nonexpansive mappings, every nonexpansive mapping of C into itself has a fixed point in C. By Theorem 1.4, C is bounded. Conversely, by Theorem 3.3, it is easy to show that (ii) ⇒ (i). □

Proof. The proof is given by induction with respect to N. We first show the case that N = 2. By Theorem 3.3, F (T1) ≠ ∅ and F (T2) ≠ ∅. Furthermore, F (T1) and F (T2) are bounded closed convex subsets of C. Furthermore, T2(F (T1)) ⊆F (T1). Indeed, if u∈F (T1), then T1T2u = T2T1u = T2u. Hence, T2u∈F (T1), and this implies that T2(F (T1)) ⊆F (T1). Let T2′:F(T1)→F(T1) be defined by T2′(x):=T2(x) for each x∈F (T1). Clearly, T2′:F(T1)→F(T1) is a asymptotic nonspreading mapping. By Theorem 3.3 again, there exists x̄∈F(T1) such that x̄=T2′(x̄)=T2(x̄). So, x̄∈F(T1)∩F(T2).

Suppose that for some n ≥ 2, X=∩k=1nF(Tk)≠∅. Then, X is a nonempty bounded closed convex subset of C. Let Tn+1′:X→Xbe defined by Tn+1′(x)=Tn+1(x) for each x∈X. Clearly, Tn+1′ is an asymptotic nonspreading mapping. By Theorem 3.3 again, we know that X ∩ F (Tn+1) ≠ ∅. That is, ∩k=1n+1F(Tk)≠∅. And the proof is completed. □

Therefore, T is an asymptotic TJ mapping. Note that T is a TJ-1 mapping. □

Remark 5.2. Example 5.2 can be applied to demonstrate Theorems 3.2, 3.3, 3.5, and Corollary 4.1. Furthermore, Examples 5.1 and 5.2 can also be applied to demonstrate Theorem 4.1.

Declarations

6 Competing interests

The authors declare no competing interests, except Prof. L. J. Lin was supported by the National Science Council of Republic of China while he work on the publish, and C. S. Chuang was support as postdoctor by the National Science Council of the Republic of China while he worked on this problem.

7 Authors' contributions

LJL: Problem resign, coordinator, discussion, revise the important part, and submit CSC: Responsible for the important results of asymptotic nonspreading mappings and asymptotic TJ mapping, discuss, draft. ZTY: responsible for giving the examples of this types of problems, discussion.

Authors’ Affiliations

(1)

Department of Mathematics, National Changhua University of Education

(2)

Department of Electronic Engineering, Nan Kai University of Technology

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